BIMSA > Seminar on Control Theory and Nonlinear Filtering Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula
Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula
Organizer
Shing Toung Yau
Speaker
Zeju Sun
Time
Wednesday, November 6, 2024 9:00 PM - 10:00 PM
Venue
Online
Abstract
In this talk, I will review a paper by X. Fang, Q.-M. Shao and L. Xu. The abstract of this paper is as follows: ‘Stein’s method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut’s formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein’s exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance. We apply the theorem to the unadjusted Langevin algorithm.